/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 873 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 2 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 132 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 3005 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 6 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 257 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 79 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 213 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 15 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 278 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 905 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 226 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 169 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 12 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 3 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 230 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 404 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 129 ms] (62) CpxRNTS (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 228 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (68) CpxRNTS (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 178 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] (74) CpxRNTS (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 238 ms] (78) CpxRNTS (79) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (80) CpxRNTS (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 289 ms] (84) CpxRNTS (85) IntTrsBoundProof [UPPER BOUND(ID), 24 ms] (86) CpxRNTS (87) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 744 ms] (90) CpxRNTS (91) IntTrsBoundProof [UPPER BOUND(ID), 166 ms] (92) CpxRNTS (93) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 106 ms] (96) CpxRNTS (97) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (98) CpxRNTS (99) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 576 ms] (102) CpxRNTS (103) IntTrsBoundProof [UPPER BOUND(ID), 137 ms] (104) CpxRNTS (105) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (106) CpxRNTS (107) IntTrsBoundProof [UPPER BOUND(ID), 1888 ms] (108) CpxRNTS (109) IntTrsBoundProof [UPPER BOUND(ID), 352 ms] (110) CpxRNTS (111) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (112) CpxRNTS (113) IntTrsBoundProof [UPPER BOUND(ID), 249 ms] (114) CpxRNTS (115) IntTrsBoundProof [UPPER BOUND(ID), 64 ms] (116) CpxRNTS (117) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (118) CpxRNTS (119) IntTrsBoundProof [UPPER BOUND(ID), 239 ms] (120) CpxRNTS (121) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (122) CpxRNTS (123) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (124) CpxRNTS (125) IntTrsBoundProof [UPPER BOUND(ID), 207 ms] (126) CpxRNTS (127) IntTrsBoundProof [UPPER BOUND(ID), 61 ms] (128) CpxRNTS (129) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (130) CpxRNTS (131) IntTrsBoundProof [UPPER BOUND(ID), 127 ms] (132) CpxRNTS (133) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (134) CpxRNTS (135) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (136) CpxRNTS (137) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] (138) CpxRNTS (139) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (140) CpxRNTS (141) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (142) CpxRNTS (143) IntTrsBoundProof [UPPER BOUND(ID), 219 ms] (144) CpxRNTS (145) IntTrsBoundProof [UPPER BOUND(ID), 92 ms] (146) CpxRNTS (147) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (148) CpxRNTS (149) IntTrsBoundProof [UPPER BOUND(ID), 349 ms] (150) CpxRNTS (151) IntTrsBoundProof [UPPER BOUND(ID), 103 ms] (152) CpxRNTS (153) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (154) CpxRNTS (155) IntTrsBoundProof [UPPER BOUND(ID), 540 ms] (156) CpxRNTS (157) IntTrsBoundProof [UPPER BOUND(ID), 164 ms] (158) CpxRNTS (159) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (160) CpxRNTS (161) IntTrsBoundProof [UPPER BOUND(ID), 641 ms] (162) CpxRNTS (163) IntTrsBoundProof [UPPER BOUND(ID), 114 ms] (164) CpxRNTS (165) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (166) CpxRNTS (167) IntTrsBoundProof [UPPER BOUND(ID), 3322 ms] (168) CpxRNTS (169) IntTrsBoundProof [UPPER BOUND(ID), 240 ms] (170) CpxRNTS (171) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (172) CpxRNTS (173) IntTrsBoundProof [UPPER BOUND(ID), 796 ms] (174) CpxRNTS (175) IntTrsBoundProof [UPPER BOUND(ID), 71 ms] (176) CpxRNTS (177) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (178) CpxRNTS (179) IntTrsBoundProof [UPPER BOUND(ID), 1209 ms] (180) CpxRNTS (181) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (182) CpxRNTS (183) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (184) CpxRNTS (185) IntTrsBoundProof [UPPER BOUND(ID), 578 ms] (186) CpxRNTS (187) IntTrsBoundProof [UPPER BOUND(ID), 72 ms] (188) CpxRNTS (189) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (190) CpxRNTS (191) IntTrsBoundProof [UPPER BOUND(ID), 756 ms] (192) CpxRNTS (193) IntTrsBoundProof [UPPER BOUND(ID), 154 ms] (194) CpxRNTS (195) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (196) CpxRNTS (197) IntTrsBoundProof [UPPER BOUND(ID), 705 ms] (198) CpxRNTS (199) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (200) CpxRNTS (201) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (202) CpxRNTS (203) IntTrsBoundProof [UPPER BOUND(ID), 675 ms] (204) CpxRNTS (205) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (206) CpxRNTS (207) FinalProof [FINISHED, 0 ms] (208) BOUNDS(1, n^2) (209) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (210) TRS for Loop Detection (211) DecreasingLoopProof [LOWER BOUND(ID), 54 ms] (212) BEST (213) proven lower bound (214) LowerBoundPropagationProof [FINISHED, 0 ms] (215) BOUNDS(n^1, INF) (216) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #equal(@x, @y) -> #eq(@x, @y) #greater(@x, @y) -> #ckgt(#compare(@x, @y)) +(@x, @y) -> #add(@x, @y) firstline(@l) -> firstline#1(@l) firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) firstline#1(nil) -> nil lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) lcs#1(@m) -> lcs#2(@m) lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) lcs#2(nil) -> #abs(#0) lcs#3(::(@len, @_@1)) -> @len lcs#3(nil) -> #abs(#0) lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) lcstable#3(nil, @l2, @x) -> nil max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) max#1(#false, @a, @b) -> @b max#1(#true, @a, @b) -> @a newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) newline#1(nil, @lastline, @y) -> nil newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) newline#2(nil, @x, @xs, @y) -> nil newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) newline#6(@elem, @nl) -> ::(@elem, @nl) newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) right(@l) -> right#1(@l) right#1(::(@x, @xs)) -> @x right#1(nil) -> #abs(#0) The (relative) TRS S consists of the following rules: #add(#0, @y) -> @y #add(#neg(#s(#0)), @y) -> #pred(@y) #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) #add(#pos(#s(#0)), @y) -> #succ(@y) #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #pred(#0) -> #neg(#s(#0)) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0))) -> #0 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0) -> #pos(#s(#0)) #succ(#neg(#s(#0))) -> #0 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #equal(@x, @y) -> #eq(@x, @y) #greater(@x, @y) -> #ckgt(#compare(@x, @y)) +(@x, @y) -> #add(@x, @y) firstline(@l) -> firstline#1(@l) firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) firstline#1(nil) -> nil lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) lcs#1(@m) -> lcs#2(@m) lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) lcs#2(nil) -> #abs(#0) lcs#3(::(@len, @_@1)) -> @len lcs#3(nil) -> #abs(#0) lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) lcstable#3(nil, @l2, @x) -> nil max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) max#1(#false, @a, @b) -> @b max#1(#true, @a, @b) -> @a newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) newline#1(nil, @lastline, @y) -> nil newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) newline#2(nil, @x, @xs, @y) -> nil newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) newline#6(@elem, @nl) -> ::(@elem, @nl) newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) right(@l) -> right#1(@l) right#1(::(@x, @xs)) -> @x right#1(nil) -> #abs(#0) The (relative) TRS S consists of the following rules: #add(#0, @y) -> @y #add(#neg(#s(#0)), @y) -> #pred(@y) #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) #add(#pos(#s(#0)), @y) -> #succ(@y) #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #pred(#0) -> #neg(#s(#0)) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0))) -> #0 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0) -> #pos(#s(#0)) #succ(#neg(#s(#0))) -> #0 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [1] #equal(@x, @y) -> #eq(@x, @y) [1] #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] +(@x, @y) -> #add(@x, @y) [1] firstline(@l) -> firstline#1(@l) [1] firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] firstline#1(nil) -> nil [1] lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) [1] lcs#1(@m) -> lcs#2(@m) [1] lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] lcs#2(nil) -> #abs(#0) [1] lcs#3(::(@len, @_@1)) -> @len [1] lcs#3(nil) -> #abs(#0) [1] lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) [1] lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] lcstable#3(nil, @l2, @x) -> nil [1] max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) [1] max#1(#false, @a, @b) -> @b [1] max#1(#true, @a, @b) -> @a [1] newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] newline#1(nil, @lastline, @y) -> nil [1] newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) [1] newline#2(nil, @x, @xs, @y) -> nil [1] newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) [1] newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [1] newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [1] newline#6(@elem, @nl) -> ::(@elem, @nl) [1] newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) [1] right(@l) -> right#1(@l) [1] right#1(::(@x, @xs)) -> @x [1] right#1(nil) -> #abs(#0) [1] #add(#0, @y) -> @y [0] #add(#neg(#s(#0)), @y) -> #pred(@y) [0] #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0] #add(#pos(#s(#0)), @y) -> #succ(@y) [0] #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #pred(#0) -> #neg(#s(#0)) [0] #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] #pred(#pos(#s(#0))) -> #0 [0] #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] #succ(#0) -> #pos(#s(#0)) [0] #succ(#neg(#s(#0))) -> #0 [0] #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [1] #equal(@x, @y) -> #eq(@x, @y) [1] #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] plus(@x, @y) -> #add(@x, @y) [1] firstline(@l) -> firstline#1(@l) [1] firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] firstline#1(nil) -> nil [1] lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) [1] lcs#1(@m) -> lcs#2(@m) [1] lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] lcs#2(nil) -> #abs(#0) [1] lcs#3(::(@len, @_@1)) -> @len [1] lcs#3(nil) -> #abs(#0) [1] lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) [1] lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] lcstable#3(nil, @l2, @x) -> nil [1] max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) [1] max#1(#false, @a, @b) -> @b [1] max#1(#true, @a, @b) -> @a [1] newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] newline#1(nil, @lastline, @y) -> nil [1] newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) [1] newline#2(nil, @x, @xs, @y) -> nil [1] newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) [1] newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [1] newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [1] newline#6(@elem, @nl) -> ::(@elem, @nl) [1] newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] newline#7(#true, @belowVal, @diagVal, @rightVal) -> plus(@diagVal, #pos(#s(#0))) [1] right(@l) -> right#1(@l) [1] right#1(::(@x, @xs)) -> @x [1] right#1(nil) -> #abs(#0) [1] #add(#0, @y) -> @y [0] #add(#neg(#s(#0)), @y) -> #pred(@y) [0] #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0] #add(#pos(#s(#0)), @y) -> #succ(@y) [0] #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #pred(#0) -> #neg(#s(#0)) [0] #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] #pred(#pos(#s(#0))) -> #0 [0] #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] #succ(#0) -> #pos(#s(#0)) [0] #succ(#neg(#s(#0))) -> #0 [0] #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [1] #equal(@x, @y) -> #eq(@x, @y) [1] #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] plus(@x, @y) -> #add(@x, @y) [1] firstline(@l) -> firstline#1(@l) [1] firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] firstline#1(nil) -> nil [1] lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) [1] lcs#1(@m) -> lcs#2(@m) [1] lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] lcs#2(nil) -> #abs(#0) [1] lcs#3(::(@len, @_@1)) -> @len [1] lcs#3(nil) -> #abs(#0) [1] lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) [1] lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] lcstable#3(nil, @l2, @x) -> nil [1] max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) [1] max#1(#false, @a, @b) -> @b [1] max#1(#true, @a, @b) -> @a [1] newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] newline#1(nil, @lastline, @y) -> nil [1] newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) [1] newline#2(nil, @x, @xs, @y) -> nil [1] newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) [1] newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [1] newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [1] newline#6(@elem, @nl) -> ::(@elem, @nl) [1] newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] newline#7(#true, @belowVal, @diagVal, @rightVal) -> plus(@diagVal, #pos(#s(#0))) [1] right(@l) -> right#1(@l) [1] right#1(::(@x, @xs)) -> @x [1] right#1(nil) -> #abs(#0) [1] #add(#0, @y) -> @y [0] #add(#neg(#s(#0)), @y) -> #pred(@y) [0] #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0] #add(#pos(#s(#0)), @y) -> #succ(@y) [0] #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #pred(#0) -> #neg(#s(#0)) [0] #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] #pred(#pos(#s(#0))) -> #0 [0] #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] #succ(#0) -> #pos(#s(#0)) [0] #succ(#neg(#s(#0))) -> #0 [0] #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] The TRS has the following type information: #abs :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #0 :: #0:#neg:#pos:#s::::nil #neg :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #pos :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #s :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #equal :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #false:#true #eq :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #false:#true #greater :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #false:#true #ckgt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #EQ:#GT:#LT plus :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #add :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil firstline :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil firstline#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil :: :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil nil :: #0:#neg:#pos:#s::::nil lcs :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil lcs#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil lcstable :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil lcs#2 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil lcs#3 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil lcstable#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil lcstable#2 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil lcstable#3 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil newline :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil max :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil max#1 :: #false:#true -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #false :: #false:#true #true :: #false:#true newline#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil newline#2 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil newline#3 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil newline#4 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil right :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil newline#5 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil newline#6 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil newline#7 :: #false:#true -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil right#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #pred :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #succ :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil #and :: #false:#true -> #false:#true -> #false:#true #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: lcs_2 lcs#1_1 lcs#2_1 lcs#3_1 (c) The following functions are completely defined: lcstable_2 #greater_2 newline#7_4 #equal_2 right_1 newline_3 newline#1_3 plus_2 right#1_1 lcstable#1_2 firstline_1 #abs_1 firstline#1_1 newline#2_4 max_2 lcstable#2_3 max#1_3 newline#3_5 lcstable#3_3 newline#4_6 newline#5_6 newline#6_2 #add_2 #and_2 #ckgt_1 #compare_2 #eq_2 #pred_1 #succ_1 Due to the following rules being added: #add(v0, v1) -> null_#add [0] #and(v0, v1) -> null_#and [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] #pred(v0) -> null_#pred [0] #succ(v0) -> null_#succ [0] newline#7(v0, v1, v2, v3) -> null_newline#7 [0] newline#1(v0, v1, v2) -> null_newline#1 [0] right#1(v0) -> null_right#1 [0] lcstable#1(v0, v1) -> null_lcstable#1 [0] #abs(v0) -> null_#abs [0] firstline#1(v0) -> null_firstline#1 [0] newline#2(v0, v1, v2, v3) -> null_newline#2 [0] max#1(v0, v1, v2) -> null_max#1 [0] lcstable#3(v0, v1, v2) -> null_lcstable#3 [0] And the following fresh constants: null_#add, null_#and, null_#ckgt, null_#compare, null_#eq, null_#pred, null_#succ, null_newline#7, null_newline#1, null_right#1, null_lcstable#1, null_#abs, null_firstline#1, null_newline#2, null_max#1, null_lcstable#3 ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [1] #equal(@x, @y) -> #eq(@x, @y) [1] #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] plus(@x, @y) -> #add(@x, @y) [1] firstline(@l) -> firstline#1(@l) [1] firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] firstline#1(nil) -> nil [1] lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) [1] lcs#1(@m) -> lcs#2(@m) [1] lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] lcs#2(nil) -> #abs(#0) [1] lcs#3(::(@len, @_@1)) -> @len [1] lcs#3(nil) -> #abs(#0) [1] lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) [1] lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] lcstable#3(nil, @l2, @x) -> nil [1] max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) [1] max#1(#false, @a, @b) -> @b [1] max#1(#true, @a, @b) -> @a [1] newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] newline#1(nil, @lastline, @y) -> nil [1] newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) [1] newline#2(nil, @x, @xs, @y) -> nil [1] newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) [1] newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [1] newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [1] newline#6(@elem, @nl) -> ::(@elem, @nl) [1] newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] newline#7(#true, @belowVal, @diagVal, @rightVal) -> plus(@diagVal, #pos(#s(#0))) [1] right(@l) -> right#1(@l) [1] right#1(::(@x, @xs)) -> @x [1] right#1(nil) -> #abs(#0) [1] #add(#0, @y) -> @y [0] #add(#neg(#s(#0)), @y) -> #pred(@y) [0] #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0] #add(#pos(#s(#0)), @y) -> #succ(@y) [0] #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #pred(#0) -> #neg(#s(#0)) [0] #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] #pred(#pos(#s(#0))) -> #0 [0] #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] #succ(#0) -> #pos(#s(#0)) [0] #succ(#neg(#s(#0))) -> #0 [0] #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] #add(v0, v1) -> null_#add [0] #and(v0, v1) -> null_#and [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] #pred(v0) -> null_#pred [0] #succ(v0) -> null_#succ [0] newline#7(v0, v1, v2, v3) -> null_newline#7 [0] newline#1(v0, v1, v2) -> null_newline#1 [0] right#1(v0) -> null_right#1 [0] lcstable#1(v0, v1) -> null_lcstable#1 [0] #abs(v0) -> null_#abs [0] firstline#1(v0) -> null_firstline#1 [0] newline#2(v0, v1, v2, v3) -> null_newline#2 [0] max#1(v0, v1, v2) -> null_max#1 [0] lcstable#3(v0, v1, v2) -> null_lcstable#3 [0] The TRS has the following type information: #abs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #0 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #neg :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #pos :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #s :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #equal :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq #eq :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq #greater :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq #ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#ckgt:null_#eq #compare :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #EQ:#GT:#LT:null_#compare plus :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #add :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 firstline :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 firstline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 :: :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 nil :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcs#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcstable :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcs#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcs#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcstable#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcstable#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcstable#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 newline :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 max :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 max#1 :: #false:#true:null_#and:null_#ckgt:null_#eq -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #false :: #false:#true:null_#and:null_#ckgt:null_#eq #true :: #false:#true:null_#and:null_#ckgt:null_#eq newline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> 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#0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 right#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #pred :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #succ :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #and :: #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare null_#add :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_#and :: #false:#true:null_#and:null_#ckgt:null_#eq null_#ckgt :: #false:#true:null_#and:null_#ckgt:null_#eq null_#compare :: #EQ:#GT:#LT:null_#compare null_#eq :: #false:#true:null_#and:null_#ckgt:null_#eq null_#pred :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_#succ :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_newline#7 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_newline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_right#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_lcstable#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_#abs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_firstline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_newline#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_max#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_lcstable#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [1] #equal(@x, @y) -> #eq(@x, @y) [1] #greater(#0, #0) -> #ckgt(#EQ) [1] #greater(#0, #neg(@y')) -> #ckgt(#GT) [1] #greater(#0, #pos(@y'')) -> #ckgt(#LT) [1] #greater(#0, #s(@y1)) -> #ckgt(#LT) [1] #greater(#neg(@x'), #0) -> #ckgt(#LT) [1] #greater(#neg(@x''), #neg(@y2)) -> #ckgt(#compare(@y2, @x'')) [1] #greater(#neg(@x1), #pos(@y3)) -> #ckgt(#LT) [1] #greater(#pos(@x2), #0) -> #ckgt(#GT) [1] #greater(#pos(@x3), #neg(@y4)) -> #ckgt(#GT) [1] #greater(#pos(@x4), #pos(@y5)) -> #ckgt(#compare(@x4, @y5)) [1] #greater(#s(@x5), #0) -> #ckgt(#GT) [1] #greater(#s(@x6), #s(@y6)) -> #ckgt(#compare(@x6, @y6)) [1] #greater(@x, @y) -> #ckgt(null_#compare) [1] plus(@x, @y) -> #add(@x, @y) [1] firstline(@l) -> firstline#1(@l) [1] firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] firstline#1(nil) -> nil [1] lcs(@l1, @l2) -> lcs#1(lcstable#1(@l1, @l2)) [2] lcs#1(@m) -> lcs#2(@m) [1] lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] lcs#2(nil) -> #abs(#0) [1] lcs#3(::(@len, @_@1)) -> @len [1] lcs#3(nil) -> #abs(#0) [1] lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable#1(@xs, @l2), @l2, @x) [2] lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] lcstable#3(nil, @l2, @x) -> nil [1] max(@a, @b) -> max#1(#ckgt(#compare(@a, @b)), @a, @b) [2] max#1(#false, @a, @b) -> @b [1] max#1(#true, @a, @b) -> @a [1] newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] newline#1(nil, @lastline, @y) -> nil [1] newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline#1(@xs, @lastline', @y), @belowVal, @lastline', @x, @y) [2] newline#2(nil, @x, @xs, @y) -> nil [1] newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right#1(@nl), @belowVal, @lastline', @nl, @x, @y) [2] newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right#1(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [2] newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#eq(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [2] newline#6(@elem, @nl) -> ::(@elem, @nl) [1] newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] newline#7(#true, @belowVal, @diagVal, @rightVal) -> plus(@diagVal, #pos(#s(#0))) [1] right(@l) -> right#1(@l) [1] right#1(::(@x, @xs)) -> @x [1] right#1(nil) -> #abs(#0) [1] #add(#0, @y) -> @y [0] #add(#neg(#s(#0)), @y) -> #pred(@y) [0] #add(#neg(#s(#s(#0))), @y) -> #pred(#succ(@y)) [0] #add(#neg(#s(#s(#s(@x7)))), @y) -> #pred(#succ(#add(#pos(#s(@x7)), @y))) [0] #add(#neg(#s(#s(@x))), @y) -> #pred(null_#add) [0] #add(#pos(#s(#0)), @y) -> #succ(@y) [0] #add(#pos(#s(#s(#0))), @y) -> #succ(#succ(@y)) [0] #add(#pos(#s(#s(#s(@x8)))), @y) -> #succ(#succ(#add(#pos(#s(@x8)), @y))) [0] #add(#pos(#s(#s(@x))), @y) -> #succ(null_#add) [0] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #pred(#0) -> #neg(#s(#0)) [0] #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] #pred(#pos(#s(#0))) -> #0 [0] #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] #succ(#0) -> #pos(#s(#0)) [0] #succ(#neg(#s(#0))) -> #0 [0] #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] #add(v0, v1) -> null_#add [0] #and(v0, v1) -> null_#and [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] #pred(v0) -> null_#pred [0] #succ(v0) -> null_#succ [0] newline#7(v0, v1, v2, v3) -> null_newline#7 [0] newline#1(v0, v1, v2) -> null_newline#1 [0] right#1(v0) -> null_right#1 [0] lcstable#1(v0, v1) -> null_lcstable#1 [0] #abs(v0) -> null_#abs [0] firstline#1(v0) -> null_firstline#1 [0] newline#2(v0, v1, v2, v3) -> null_newline#2 [0] max#1(v0, v1, v2) -> null_max#1 [0] lcstable#3(v0, v1, v2) -> null_lcstable#3 [0] The TRS has the following type information: #abs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #0 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #neg :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #pos :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #s :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #equal :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq #eq :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq #greater :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq #ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#ckgt:null_#eq #compare :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #EQ:#GT:#LT:null_#compare plus :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #add :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 firstline :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 firstline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 :: :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 nil :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcs#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcstable :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcs#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcs#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcstable#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcstable#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 lcstable#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 newline :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 max :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 max#1 :: #false:#true:null_#and:null_#ckgt:null_#eq -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #false :: #false:#true:null_#and:null_#ckgt:null_#eq #true :: #false:#true:null_#and:null_#ckgt:null_#eq newline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 newline#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 newline#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 newline#4 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 right :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 newline#5 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 newline#6 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 newline#7 :: #false:#true:null_#and:null_#ckgt:null_#eq -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 right#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #pred :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #succ :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 #and :: #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare null_#add :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_#and :: #false:#true:null_#and:null_#ckgt:null_#eq null_#ckgt :: #false:#true:null_#and:null_#ckgt:null_#eq null_#compare :: #EQ:#GT:#LT:null_#compare null_#eq :: #false:#true:null_#and:null_#ckgt:null_#eq null_#pred :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_#succ :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_newline#7 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_newline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_right#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_lcstable#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_#abs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_firstline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_newline#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_max#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 null_lcstable#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: #0 => 0 nil => 1 #false => 1 #true => 2 #EQ => 1 #GT => 2 #LT => 3 null_#add => 0 null_#and => 0 null_#ckgt => 0 null_#compare => 0 null_#eq => 0 null_#pred => 0 null_#succ => 0 null_newline#7 => 0 null_newline#1 => 0 null_right#1 => 0 null_lcstable#1 => 0 null_#abs => 0 null_firstline#1 => 0 null_newline#2 => 0 null_max#1 => 0 null_lcstable#3 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x #add(z, z') -{ 0 }-> @y :|: z' = @y, z = 0, @y >= 0 #add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #add(z, z') -{ 0 }-> #succ(@y) :|: z = 1 + (1 + 0), z' = @y, @y >= 0 #add(z, z') -{ 0 }-> #succ(0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0 #add(z, z') -{ 0 }-> #succ(#succ(@y)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + @x8), @y))) :|: z = 1 + (1 + (1 + (1 + @x8))), @x8 >= 0, z' = @y, @y >= 0 #add(z, z') -{ 0 }-> #pred(@y) :|: z = 1 + (1 + 0), z' = @y, @y >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0 #add(z, z') -{ 0 }-> #pred(#succ(@y)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + @x7), @y))) :|: z' = @y, z = 1 + (1 + (1 + (1 + @x7))), @x7 >= 0, @y >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z = 1 + @x', @x' >= 0, z' = 0 #greater(z, z') -{ 1 }-> #ckgt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @y' >= 0, z' = 1 + @y', z = 0 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0 #greater(z, z') -{ 1 }-> #ckgt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0 #greater(z, z') -{ 1 }-> #ckgt(1) :|: z = 0, z' = 0 #greater(z, z') -{ 1 }-> #ckgt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) firstline(z) -{ 1 }-> firstline#1(@l) :|: z = @l, @l >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 firstline#1(z) -{ 1 }-> 1 + #abs(0) + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(@l1, @l2)) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 lcs#1(z) -{ 1 }-> lcs#2(@m) :|: @m >= 0, z = @m lcs#2(z) -{ 1 }-> lcs#3(@l1) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0 lcs#2(z) -{ 1 }-> #abs(0) :|: z = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 1 }-> #abs(0) :|: z = 1 lcstable(z, z') -{ 1 }-> lcstable#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, @l2), @l2, @x) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 lcstable#1(z, z') -{ 1 }-> 1 + firstline(@l2) + 1 :|: z' = @l2, z = 1, @l2 >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(@m, @l2, @x) :|: @m >= 0, z' = @l2, @x >= 0, @l2 >= 0, z = @m, z'' = @x lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z' = @l2, @x >= 0, z = 1, @l2 >= 0, z'' = @x lcstable#3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(@x, @l, @l2) + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' = @l2, @x >= 0, @l2 >= 0, z'' = @x max(z, z') -{ 2 }-> max#1(#ckgt(#compare(@a, @b)), @a, @b) :|: @a >= 0, z = @a, z' = @b, @b >= 0 max#1(z, z', z'') -{ 1 }-> @a :|: z = 2, @a >= 0, z' = @a, @b >= 0, z'' = @b max#1(z, z', z'') -{ 1 }-> @b :|: @a >= 0, z = 1, z' = @a, @b >= 0, z'' = @b max#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 newline(z, z', z'') -{ 1 }-> newline#1(@l, @lastline, @y) :|: @l >= 0, z'' = @l, @lastline >= 0, z' = @lastline, z = @y, @y >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(@lastline, @x, @xs, @y) :|: @x >= 0, z = 1 + @x + @xs, @lastline >= 0, z' = @lastline, @xs >= 0, @y >= 0, z'' = @y newline#1(z, z', z'') -{ 1 }-> 1 :|: @lastline >= 0, z = 1, z' = @lastline, @y >= 0, z'' = @y newline#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(@xs, @lastline', @y), @belowVal, @lastline', @x, @y) :|: @lastline' >= 0, @x >= 0, z1 = @y, z = 1 + @belowVal + @lastline', @xs >= 0, @y >= 0, z' = @x, z'' = @xs, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: @x >= 0, z = 1, z1 = @y, @xs >= 0, @y >= 0, z' = @x, z'' = @xs newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(right#1(@nl), @belowVal, @lastline', @nl, @x, @y) :|: z'' = @lastline', z1 = @x, @lastline' >= 0, @x >= 0, z = @nl, z2 = @y, z' = @belowVal, @y >= 0, @nl >= 0, @belowVal >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(right#1(@lastline'), @belowVal, @nl, @rightVal, @x, @y) :|: z'' = @lastline', @lastline' >= 0, @x >= 0, @rightVal >= 0, z1 = @nl, z2 = @x, z = @rightVal, z' = @belowVal, z3 = @y, @y >= 0, @belowVal >= 0, @nl >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(@x, @y), @belowVal, @diagVal, @rightVal), @nl) :|: z = @diagVal, @x >= 0, @rightVal >= 0, z2 = @x, z' = @belowVal, z3 = @y, z'' = @nl, z1 = @rightVal, @diagVal >= 0, @y >= 0, @belowVal >= 0, @nl >= 0 newline#6(z, z') -{ 1 }-> 1 + @elem + @nl :|: @elem >= 0, z' = @nl, @nl >= 0, z = @elem newline#7(z, z', z'', z1) -{ 1 }-> plus(@diagVal, 1 + (1 + 0)) :|: z = 2, @rightVal >= 0, z' = @belowVal, z1 = @rightVal, @diagVal >= 0, z'' = @diagVal, @belowVal >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(@belowVal, @rightVal) :|: @rightVal >= 0, z = 1, z' = @belowVal, z1 = @rightVal, @diagVal >= 0, z'' = @diagVal, @belowVal >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 plus(z, z') -{ 1 }-> #add(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 right(z) -{ 1 }-> right#1(@l) :|: z = @l, @l >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 right#1(z) -{ 1 }-> #abs(0) :|: z = 1 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 max#1(z, z', z'') -{ 1 }-> @b :|: @a >= 0, z = 1, z' = @a, @b >= 0, z'' = @b max#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 max#1(z, z', z'') -{ 1 }-> @a :|: z = 2, @a >= 0, z' = @a, @b >= 0, z'' = @b newline#6(z, z') -{ 1 }-> 1 + @elem + @nl :|: @elem >= 0, z' = @nl, @nl >= 0, z = @elem #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 lcs#3(z) -{ 1 }-> #abs(0) :|: z = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 right#1(z) -{ 1 }-> #abs(0) :|: z = 1 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x #add(z, z') -{ 0 }-> @y :|: z' = @y, z = 0, @y >= 0 #add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @y = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' = @y, @y >= 0, v0 >= 0, @y = v0 #add(z, z') -{ 0 }-> 0 :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), v0 >= 0, 1 + (1 + @x) = v0 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), 1 + (1 + @x) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), v0 >= 0, 1 + (1 + (1 + @x)) = v0 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), @x' >= 0, 1 + (1 + @x) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), @x' >= 0, 1 + (1 + (1 + @x)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @y = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @x >= 0, @y = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), @x' >= 0, 1 + (1 + @x) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), @x' >= 0, 1 + (1 + (1 + @x)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + @x8), @y))) :|: z = 1 + (1 + (1 + (1 + @x8))), @x8 >= 0, z' = @y, @y >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0 #add(z, z') -{ 0 }-> #pred(0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + @x)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + @x))) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x) #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + @x7), @y))) :|: z' = @y, z = 1 + (1 + (1 + (1 + @x7))), @x7 >= 0, @y >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #greater(z, z') -{ 1 }-> 2 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) firstline(z) -{ 1 }-> firstline#1(@l) :|: z = @l, @l >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(@l1, @l2)) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 lcs#1(z) -{ 1 }-> lcs#2(@m) :|: @m >= 0, z = @m lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, @l2), @l2, @x) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 lcstable#1(z, z') -{ 1 }-> 1 + firstline(@l2) + 1 :|: z' = @l2, z = 1, @l2 >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(@m, @l2, @x) :|: @m >= 0, z' = @l2, @x >= 0, @l2 >= 0, z = @m, z'' = @x lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z' = @l2, @x >= 0, z = 1, @l2 >= 0, z'' = @x lcstable#3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(@x, @l, @l2) + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' = @l2, @x >= 0, @l2 >= 0, z'' = @x max(z, z') -{ 2 }-> max#1(#ckgt(#compare(@a, @b)), @a, @b) :|: @a >= 0, z = @a, z' = @b, @b >= 0 max#1(z, z', z'') -{ 1 }-> @a :|: z = 2, @a >= 0, z' = @a, @b >= 0, z'' = @b max#1(z, z', z'') -{ 1 }-> @b :|: @a >= 0, z = 1, z' = @a, @b >= 0, z'' = @b max#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 newline(z, z', z'') -{ 1 }-> newline#1(@l, @lastline, @y) :|: @l >= 0, z'' = @l, @lastline >= 0, z' = @lastline, z = @y, @y >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(@lastline, @x, @xs, @y) :|: @x >= 0, z = 1 + @x + @xs, @lastline >= 0, z' = @lastline, @xs >= 0, @y >= 0, z'' = @y newline#1(z, z', z'') -{ 1 }-> 1 :|: @lastline >= 0, z = 1, z' = @lastline, @y >= 0, z'' = @y newline#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(@xs, @lastline', @y), @belowVal, @lastline', @x, @y) :|: @lastline' >= 0, @x >= 0, z1 = @y, z = 1 + @belowVal + @lastline', @xs >= 0, @y >= 0, z' = @x, z'' = @xs, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: @x >= 0, z = 1, z1 = @y, @xs >= 0, @y >= 0, z' = @x, z'' = @xs newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', @belowVal, @lastline', @nl, @x, @y) :|: z'' = @lastline', z1 = @x, @lastline' >= 0, @x >= 0, z = @nl, z2 = @y, z' = @belowVal, @y >= 0, @nl >= 0, @belowVal >= 0, @x' >= 0, @nl = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, @belowVal, @lastline', @nl, @x, @y) :|: z'' = @lastline', z1 = @x, @lastline' >= 0, @x >= 0, z = @nl, z2 = @y, z' = @belowVal, @y >= 0, @nl >= 0, @belowVal >= 0, v0 >= 0, @nl = v0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), @belowVal, @lastline', @nl, @x, @y) :|: z'' = @lastline', z1 = @x, @lastline' >= 0, @x >= 0, z = @nl, z2 = @y, z' = @belowVal, @y >= 0, @nl >= 0, @belowVal >= 0, @nl = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', @belowVal, @nl, @rightVal, @x, @y) :|: z'' = @lastline', @lastline' >= 0, @x >= 0, @rightVal >= 0, z1 = @nl, z2 = @x, z = @rightVal, z' = @belowVal, z3 = @y, @y >= 0, @belowVal >= 0, @nl >= 0, @x' >= 0, @lastline' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, @belowVal, @nl, @rightVal, @x, @y) :|: z'' = @lastline', @lastline' >= 0, @x >= 0, @rightVal >= 0, z1 = @nl, z2 = @x, z = @rightVal, z' = @belowVal, z3 = @y, @y >= 0, @belowVal >= 0, @nl >= 0, v0 >= 0, @lastline' = v0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), @belowVal, @nl, @rightVal, @x, @y) :|: z'' = @lastline', @lastline' >= 0, @x >= 0, @rightVal >= 0, z1 = @nl, z2 = @x, z = @rightVal, z' = @belowVal, z3 = @y, @y >= 0, @belowVal >= 0, @nl >= 0, @lastline' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(@x, @y), @belowVal, @diagVal, @rightVal), @nl) :|: z = @diagVal, @x >= 0, @rightVal >= 0, z2 = @x, z' = @belowVal, z3 = @y, z'' = @nl, z1 = @rightVal, @diagVal >= 0, @y >= 0, @belowVal >= 0, @nl >= 0 newline#6(z, z') -{ 1 }-> 1 + @elem + @nl :|: @elem >= 0, z' = @nl, @nl >= 0, z = @elem newline#7(z, z', z'', z1) -{ 1 }-> plus(@diagVal, 1 + (1 + 0)) :|: z = 2, @rightVal >= 0, z' = @belowVal, z1 = @rightVal, @diagVal >= 0, z'' = @diagVal, @belowVal >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(@belowVal, @rightVal) :|: @rightVal >= 0, z = 1, z' = @belowVal, z1 = @rightVal, @diagVal >= 0, z'' = @diagVal, @belowVal >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 plus(z, z') -{ 1 }-> #add(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 right(z) -{ 2 }-> @x :|: z = @l, @l >= 0, @x >= 0, @l = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z = @l, @l >= 0, v0 >= 0, @l = v0 right(z) -{ 2 }-> #abs(0) :|: z = @l, @l >= 0, @l = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { #and } { right#1 } { max#1 } { #compare } { #ckgt } { newline#6 } { firstline#1, firstline } { #pred } { lcs#3 } { #abs } { #succ } { #eq } { max } { #greater } { #add } { right } { lcs#2 } { #equal } { plus } { lcs#1 } { newline#7 } { newline#5 } { newline#4 } { newline#3 } { newline#1, newline#2 } { newline } { lcstable#3 } { lcstable#2 } { lcstable#1 } { lcstable } { lcs } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#and}, {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#and}, {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#and}, {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: ?, size: O(1) [2] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: right#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: right#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: max#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: max#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: ?, size: O(1) [3] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(s2), z, z') :|: s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #ckgt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(s2), z, z') :|: s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: ?, size: O(1) [2] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #ckgt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 2 }-> max#1(#ckgt(s2), z, z') :|: s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: newline#6 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: newline#6 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] ---------------------------------------- (57) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: firstline#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z Computed SIZE bound using CoFloCo for: firstline after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: ?, size: O(n^1) [z] firstline: runtime: ?, size: O(n^1) [z] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: firstline#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + 3*z Computed RUNTIME bound using CoFloCo for: firstline after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + 3*z ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] ---------------------------------------- (63) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #pred after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: ?, size: O(n^1) [2 + z] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #pred after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] ---------------------------------------- (69) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: lcs#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: ?, size: O(n^1) [z] ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lcs#3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] ---------------------------------------- (75) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #abs after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (79) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #abs after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 1 }-> 0 :|: z >= 0 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (81) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #succ after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: ?, size: O(n^1) [2 + z] ---------------------------------------- (85) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #succ after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] ---------------------------------------- (87) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: ?, size: O(1) [2] ---------------------------------------- (91) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (93) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: max after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (97) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: max after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] ---------------------------------------- (99) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #greater after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: ?, size: O(1) [2] ---------------------------------------- (103) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #greater after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (104) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (105) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (106) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (107) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: #add after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z + z' ---------------------------------------- (108) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: ?, size: O(n^1) [2*z + z'] ---------------------------------------- (109) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #add after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (110) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] ---------------------------------------- (111) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (112) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] ---------------------------------------- (113) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: right after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (114) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: ?, size: O(n^1) [z] ---------------------------------------- (115) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: right after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (116) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] ---------------------------------------- (117) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (118) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] ---------------------------------------- (119) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: lcs#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (120) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: ?, size: O(n^1) [z] ---------------------------------------- (121) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lcs#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (122) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] ---------------------------------------- (123) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (124) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] ---------------------------------------- (125) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #equal after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (126) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: ?, size: O(1) [2] ---------------------------------------- (127) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #equal after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (128) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (129) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (130) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (131) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z + z' ---------------------------------------- (132) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: ?, size: O(n^1) [2*z + z'] ---------------------------------------- (133) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (134) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] ---------------------------------------- (135) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (136) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] ---------------------------------------- (137) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lcs#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (138) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (139) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lcs#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 ---------------------------------------- (140) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] ---------------------------------------- (141) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (142) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] ---------------------------------------- (143) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: newline#7 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + 2*z'' + z1 ---------------------------------------- (144) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: ?, size: O(n^1) [2 + z' + 2*z'' + z1] ---------------------------------------- (145) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: newline#7 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 ---------------------------------------- (146) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] ---------------------------------------- (147) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (148) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] ---------------------------------------- (149) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: newline#5 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z + z' + z'' + z1 ---------------------------------------- (150) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: ?, size: O(n^1) [3 + 2*z + z' + z'' + z1] ---------------------------------------- (151) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: newline#5 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 7 ---------------------------------------- (152) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] ---------------------------------------- (153) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (154) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] ---------------------------------------- (155) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: newline#4 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z + z' + 2*z'' + z1 ---------------------------------------- (156) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: ?, size: O(n^1) [3 + z + z' + 2*z'' + z1] ---------------------------------------- (157) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: newline#4 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 11 ---------------------------------------- (158) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] ---------------------------------------- (159) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (160) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] ---------------------------------------- (161) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: newline#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z + z' + 2*z'' ---------------------------------------- (162) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: ?, size: O(n^1) [3 + 2*z + z' + 2*z''] ---------------------------------------- (163) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: newline#3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 15 ---------------------------------------- (164) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] ---------------------------------------- (165) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (166) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] ---------------------------------------- (167) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: newline#1 after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? Computed SIZE bound using CoFloCo for: newline#2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (168) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: ?, size: EXP newline#2: runtime: ?, size: INF ---------------------------------------- (169) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: newline#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 20 + 18*z Computed RUNTIME bound using CoFloCo for: newline#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 37 + 18*z'' ---------------------------------------- (170) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF ---------------------------------------- (171) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (172) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF ---------------------------------------- (173) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: newline after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (174) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: ?, size: INF ---------------------------------------- (175) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: newline after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 21 + 18*z'' ---------------------------------------- (176) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF ---------------------------------------- (177) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (178) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF ---------------------------------------- (179) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lcstable#3 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (180) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: ?, size: INF ---------------------------------------- (181) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lcstable#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 22 + 18*z' ---------------------------------------- (182) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF ---------------------------------------- (183) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (184) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF ---------------------------------------- (185) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lcstable#2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (186) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: ?, size: INF ---------------------------------------- (187) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lcstable#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 23 + 18*z' ---------------------------------------- (188) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF ---------------------------------------- (189) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (190) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF ---------------------------------------- (191) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lcstable#1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (192) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable#1}, {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF lcstable#1: runtime: ?, size: INF ---------------------------------------- (193) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: lcstable#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 7 + 25*z + 18*z*z' + 3*z' ---------------------------------------- (194) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF ---------------------------------------- (195) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (196) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF ---------------------------------------- (197) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lcstable after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (198) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcstable}, {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF lcstable: runtime: ?, size: INF ---------------------------------------- (199) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: lcstable after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 8 + 25*z + 18*z*z' + 3*z' ---------------------------------------- (200) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF lcstable: runtime: O(n^2) [8 + 25*z + 18*z*z' + 3*z'], size: INF ---------------------------------------- (201) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (202) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF lcstable: runtime: O(n^2) [8 + 25*z + 18*z*z' + 3*z'], size: INF ---------------------------------------- (203) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lcs after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (204) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: {lcs} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF lcstable: runtime: O(n^2) [8 + 25*z + 18*z*z' + 3*z'], size: INF lcs: runtime: ?, size: INF ---------------------------------------- (205) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: lcs after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 13 + 25*z + 18*z*z' + 3*z' ---------------------------------------- (206) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #pred(z) -{ 0 }-> 0 :|: z >= 0 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) #succ(z) -{ 0 }-> 0 :|: z >= 0 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 firstline#1(z) -{ 1 }-> 1 :|: z = 1 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 right(z) -{ 1 }-> 0 :|: z >= 0 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 right#1(z) -{ 0 }-> 0 :|: z >= 0 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 Function symbols to be analyzed: Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] right#1: runtime: O(1) [2], size: O(n^1) [z] max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] #pred: runtime: O(1) [0], size: O(n^1) [2 + z] lcs#3: runtime: O(1) [2], size: O(n^1) [z] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #succ: runtime: O(1) [0], size: O(n^1) [2 + z] #eq: runtime: O(1) [0], size: O(1) [2] max: runtime: O(1) [3], size: O(n^1) [z + z'] #greater: runtime: O(1) [1], size: O(1) [2] #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] right: runtime: O(1) [3], size: O(n^1) [z] lcs#2: runtime: O(1) [3], size: O(n^1) [z] #equal: runtime: O(1) [1], size: O(1) [2] plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] lcs#1: runtime: O(1) [4], size: O(n^1) [z] newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] newline#1: runtime: O(n^1) [20 + 18*z], size: EXP newline#2: runtime: O(n^1) [37 + 18*z''], size: INF newline: runtime: O(n^1) [21 + 18*z''], size: INF lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF lcstable: runtime: O(n^2) [8 + 25*z + 18*z*z' + 3*z'], size: INF lcs: runtime: O(n^2) [13 + 25*z + 18*z*z' + 3*z'], size: INF ---------------------------------------- (207) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (208) BOUNDS(1, n^2) ---------------------------------------- (209) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (210) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #equal(@x, @y) -> #eq(@x, @y) #greater(@x, @y) -> #ckgt(#compare(@x, @y)) +(@x, @y) -> #add(@x, @y) firstline(@l) -> firstline#1(@l) firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) firstline#1(nil) -> nil lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) lcs#1(@m) -> lcs#2(@m) lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) lcs#2(nil) -> #abs(#0) lcs#3(::(@len, @_@1)) -> @len lcs#3(nil) -> #abs(#0) lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) lcstable#3(nil, @l2, @x) -> nil max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) max#1(#false, @a, @b) -> @b max#1(#true, @a, @b) -> @a newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) newline#1(nil, @lastline, @y) -> nil newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) newline#2(nil, @x, @xs, @y) -> nil newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) newline#6(@elem, @nl) -> ::(@elem, @nl) newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) right(@l) -> right#1(@l) right#1(::(@x, @xs)) -> @x right#1(nil) -> #abs(#0) The (relative) TRS S consists of the following rules: #add(#0, @y) -> @y #add(#neg(#s(#0)), @y) -> #pred(@y) #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) #add(#pos(#s(#0)), @y) -> #succ(@y) #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #pred(#0) -> #neg(#s(#0)) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0))) -> #0 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0) -> #pos(#s(#0)) #succ(#neg(#s(#0))) -> #0 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) Rewrite Strategy: INNERMOST ---------------------------------------- (211) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence firstline#1(::(@x, @xs)) ->^+ ::(#abs(#0), firstline#1(@xs)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [@xs / ::(@x, @xs)]. The result substitution is [ ]. ---------------------------------------- (212) Complex Obligation (BEST) ---------------------------------------- (213) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #equal(@x, @y) -> #eq(@x, @y) #greater(@x, @y) -> #ckgt(#compare(@x, @y)) +(@x, @y) -> #add(@x, @y) firstline(@l) -> firstline#1(@l) firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) firstline#1(nil) -> nil lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) lcs#1(@m) -> lcs#2(@m) lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) lcs#2(nil) -> #abs(#0) lcs#3(::(@len, @_@1)) -> @len lcs#3(nil) -> #abs(#0) lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) lcstable#3(nil, @l2, @x) -> nil max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) max#1(#false, @a, @b) -> @b max#1(#true, @a, @b) -> @a newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) newline#1(nil, @lastline, @y) -> nil newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) newline#2(nil, @x, @xs, @y) -> nil newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) newline#6(@elem, @nl) -> ::(@elem, @nl) newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) right(@l) -> right#1(@l) right#1(::(@x, @xs)) -> @x right#1(nil) -> #abs(#0) The (relative) TRS S consists of the following rules: #add(#0, @y) -> @y #add(#neg(#s(#0)), @y) -> #pred(@y) #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) #add(#pos(#s(#0)), @y) -> #succ(@y) #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #pred(#0) -> #neg(#s(#0)) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0))) -> #0 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0) -> #pos(#s(#0)) #succ(#neg(#s(#0))) -> #0 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) Rewrite Strategy: INNERMOST ---------------------------------------- (214) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (215) BOUNDS(n^1, INF) ---------------------------------------- (216) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #equal(@x, @y) -> #eq(@x, @y) #greater(@x, @y) -> #ckgt(#compare(@x, @y)) +(@x, @y) -> #add(@x, @y) firstline(@l) -> firstline#1(@l) firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) firstline#1(nil) -> nil lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) lcs#1(@m) -> lcs#2(@m) lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) lcs#2(nil) -> #abs(#0) lcs#3(::(@len, @_@1)) -> @len lcs#3(nil) -> #abs(#0) lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) lcstable#3(nil, @l2, @x) -> nil max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) max#1(#false, @a, @b) -> @b max#1(#true, @a, @b) -> @a newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) newline#1(nil, @lastline, @y) -> nil newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) newline#2(nil, @x, @xs, @y) -> nil newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) newline#6(@elem, @nl) -> ::(@elem, @nl) newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) right(@l) -> right#1(@l) right#1(::(@x, @xs)) -> @x right#1(nil) -> #abs(#0) The (relative) TRS S consists of the following rules: #add(#0, @y) -> @y #add(#neg(#s(#0)), @y) -> #pred(@y) #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) #add(#pos(#s(#0)), @y) -> #succ(@y) #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #pred(#0) -> #neg(#s(#0)) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0))) -> #0 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0) -> #pos(#s(#0)) #succ(#neg(#s(#0))) -> #0 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) Rewrite Strategy: INNERMOST